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  1. We propose and study a nonlinear elimination preconditioned inexact Newton method for the numerical simulation of diseased human arteries with a heterogeneous hyperelastic model. We assume the artery is made of layers of distinct tissues and also contains plaque. Traditional Newton methods often work well for smooth and homogeneous arteries but suffer from slow or no convergence due to the heterogeneousness of diseased soft tissues when the material is quasi-incompressible. The proposed nonlinear elimination method adaptively finds a small number of equations causing the nonlinear stagnation and then eliminates them from the global nonlinear system. By using the theory ofmore »affine invariance of Newton method, we provide insight into why the nonlinear elimination method can improve the convergence of Newton iterations. Our numerical results show that the combination of nonlinear elimination with an initial guess interpolated from a coarse level solution can lead to the uniform convergence of Newton method for this class of very difficult nonlinear problems.« less
  2. Mainstream random walks on graphs mostly focus on the topology while ignoring node attributes. In this paper, we develop a matrix form of the attributed random walk with pointwise mutual information in an unsupervised fashion. We show through experiments that the generated embeddings of flexible dimensions are robust to label missing on the transductive node classification task.
  3. Variable selection plays a fundamental role in high-dimensional data analysis. Various methods have been developed for variable selection in recent years. Well-known examples are forward stepwise regression (FSR) and least angle regression (LARS), among others. These methods typically add variables into the model one by one. For such selection procedures, it is crucial to find a stopping criterion that controls model complexity. One of the most commonly used techniques to this end is cross-validation (CV) which, in spite of its popularity, has two major drawbacks: expensive computational cost and lack of statistical interpretation. To overcome these drawbacks, we introduce amore »flexible and efficient test-based variable selection approach that can be incorporated into any sequential selection procedure. The test, which is on the overall signal in the remaining inactive variables, is based on the maximal absolute partial correlation between the inactive variables and the response given active variables. We develop the asymptotic null distribution of the proposed test statistic as the dimension tends to infinity uniformly in the sample size. We also show that the test is consistent. With this test, at each step of the selection, a new variable is included if and only if the -value is below some pre-defined level. Numerical studies show that the proposed method delivers very competitive performance in terms of variable selection accuracy and computational complexity compared to CV.« less